2.4.1 Largest Eigenvalue and Related Eigenvector

Regard the $N$ eigenvectors $\vec{a}_{i}$ of $A$
as the base vectors in $N$-space. Then any
$N$-vector may be written

$
\vec{x}_{0}= \sum \limits_{i=1}^{N} c_{i} \vec{a}_{i}
$

with appropriate coefficients
$c_{i}$. Let $\vec{a}_{m}$ be the eigenvector corresponding to the largest
(by absolute value) eigenvalue $\lambda_{m}$. Multiply $\vec{x}_{0}$
several times by ${ A}$, each time normalizing the result: